Optimal. Leaf size=166 \[ \frac {256 c^3 (b+2 c x) (3 b B-4 A c)}{63 b^7 \sqrt {b x+c x^2}}-\frac {32 c^2 (b+2 c x) (3 b B-4 A c)}{63 b^5 \left (b x+c x^2\right )^{3/2}}+\frac {4 c (3 b B-4 A c)}{21 b^3 x \left (b x+c x^2\right )^{3/2}}-\frac {2 (3 b B-4 A c)}{21 b^2 x^2 \left (b x+c x^2\right )^{3/2}}-\frac {2 A}{9 b x^3 \left (b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.15, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {792, 658, 614, 613} \begin {gather*} \frac {256 c^3 (b+2 c x) (3 b B-4 A c)}{63 b^7 \sqrt {b x+c x^2}}-\frac {32 c^2 (b+2 c x) (3 b B-4 A c)}{63 b^5 \left (b x+c x^2\right )^{3/2}}+\frac {4 c (3 b B-4 A c)}{21 b^3 x \left (b x+c x^2\right )^{3/2}}-\frac {2 (3 b B-4 A c)}{21 b^2 x^2 \left (b x+c x^2\right )^{3/2}}-\frac {2 A}{9 b x^3 \left (b x+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 613
Rule 614
Rule 658
Rule 792
Rubi steps
\begin {align*} \int \frac {A+B x}{x^3 \left (b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 A}{9 b x^3 \left (b x+c x^2\right )^{3/2}}--\frac {\left (2 \left (-3 (-b B+A c)-\frac {3}{2} (-b B+2 A c)\right )\right ) \int \frac {1}{x^2 \left (b x+c x^2\right )^{5/2}} \, dx}{9 b}\\ &=-\frac {2 A}{9 b x^3 \left (b x+c x^2\right )^{3/2}}-\frac {2 (3 b B-4 A c)}{21 b^2 x^2 \left (b x+c x^2\right )^{3/2}}-\frac {(10 c (3 b B-4 A c)) \int \frac {1}{x \left (b x+c x^2\right )^{5/2}} \, dx}{21 b^2}\\ &=-\frac {2 A}{9 b x^3 \left (b x+c x^2\right )^{3/2}}-\frac {2 (3 b B-4 A c)}{21 b^2 x^2 \left (b x+c x^2\right )^{3/2}}+\frac {4 c (3 b B-4 A c)}{21 b^3 x \left (b x+c x^2\right )^{3/2}}+\frac {\left (16 c^2 (3 b B-4 A c)\right ) \int \frac {1}{\left (b x+c x^2\right )^{5/2}} \, dx}{21 b^3}\\ &=-\frac {2 A}{9 b x^3 \left (b x+c x^2\right )^{3/2}}-\frac {2 (3 b B-4 A c)}{21 b^2 x^2 \left (b x+c x^2\right )^{3/2}}+\frac {4 c (3 b B-4 A c)}{21 b^3 x \left (b x+c x^2\right )^{3/2}}-\frac {32 c^2 (3 b B-4 A c) (b+2 c x)}{63 b^5 \left (b x+c x^2\right )^{3/2}}-\frac {\left (128 c^3 (3 b B-4 A c)\right ) \int \frac {1}{\left (b x+c x^2\right )^{3/2}} \, dx}{63 b^5}\\ &=-\frac {2 A}{9 b x^3 \left (b x+c x^2\right )^{3/2}}-\frac {2 (3 b B-4 A c)}{21 b^2 x^2 \left (b x+c x^2\right )^{3/2}}+\frac {4 c (3 b B-4 A c)}{21 b^3 x \left (b x+c x^2\right )^{3/2}}-\frac {32 c^2 (3 b B-4 A c) (b+2 c x)}{63 b^5 \left (b x+c x^2\right )^{3/2}}+\frac {256 c^3 (3 b B-4 A c) (b+2 c x)}{63 b^7 \sqrt {b x+c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 145, normalized size = 0.87 \begin {gather*} \frac {6 b B x \left (-3 b^5+6 b^4 c x-16 b^3 c^2 x^2+96 b^2 c^3 x^3+384 b c^4 x^4+256 c^5 x^5\right )-2 A \left (7 b^6-12 b^5 c x+24 b^4 c^2 x^2-64 b^3 c^3 x^3+384 b^2 c^4 x^4+1536 b c^5 x^5+1024 c^6 x^6\right )}{63 b^7 x^3 (x (b+c x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.48, size = 163, normalized size = 0.98 \begin {gather*} \frac {2 \sqrt {b x+c x^2} \left (-7 A b^6+12 A b^5 c x-24 A b^4 c^2 x^2+64 A b^3 c^3 x^3-384 A b^2 c^4 x^4-1536 A b c^5 x^5-1024 A c^6 x^6-9 b^6 B x+18 b^5 B c x^2-48 b^4 B c^2 x^3+288 b^3 B c^3 x^4+1152 b^2 B c^4 x^5+768 b B c^5 x^6\right )}{63 b^7 x^5 (b+c x)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 177, normalized size = 1.07 \begin {gather*} -\frac {2 \, {\left (7 \, A b^{6} - 256 \, {\left (3 \, B b c^{5} - 4 \, A c^{6}\right )} x^{6} - 384 \, {\left (3 \, B b^{2} c^{4} - 4 \, A b c^{5}\right )} x^{5} - 96 \, {\left (3 \, B b^{3} c^{3} - 4 \, A b^{2} c^{4}\right )} x^{4} + 16 \, {\left (3 \, B b^{4} c^{2} - 4 \, A b^{3} c^{3}\right )} x^{3} - 6 \, {\left (3 \, B b^{5} c - 4 \, A b^{4} c^{2}\right )} x^{2} + 3 \, {\left (3 \, B b^{6} - 4 \, A b^{5} c\right )} x\right )} \sqrt {c x^{2} + b x}}{63 \, {\left (b^{7} c^{2} x^{7} + 2 \, b^{8} c x^{6} + b^{9} x^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {B x + A}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}} x^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 158, normalized size = 0.95 \begin {gather*} -\frac {2 \left (c x +b \right ) \left (1024 A \,c^{6} x^{6}-768 B b \,c^{5} x^{6}+1536 A b \,c^{5} x^{5}-1152 B \,b^{2} c^{4} x^{5}+384 A \,b^{2} c^{4} x^{4}-288 B \,b^{3} c^{3} x^{4}-64 A \,b^{3} c^{3} x^{3}+48 B \,b^{4} c^{2} x^{3}+24 A \,b^{4} c^{2} x^{2}-18 B \,b^{5} c \,x^{2}-12 A \,b^{5} c x +9 b^{6} B x +7 A \,b^{6}\right )}{63 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} b^{7} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 270, normalized size = 1.63 \begin {gather*} -\frac {64 \, B c^{3} x}{21 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{4}} + \frac {512 \, B c^{4} x}{21 \, \sqrt {c x^{2} + b x} b^{6}} + \frac {256 \, A c^{4} x}{63 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{5}} - \frac {2048 \, A c^{5} x}{63 \, \sqrt {c x^{2} + b x} b^{7}} - \frac {32 \, B c^{2}}{21 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{3}} + \frac {256 \, B c^{3}}{21 \, \sqrt {c x^{2} + b x} b^{5}} + \frac {128 \, A c^{3}}{63 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{4}} - \frac {1024 \, A c^{4}}{63 \, \sqrt {c x^{2} + b x} b^{6}} + \frac {4 \, B c}{7 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2} x} - \frac {16 \, A c^{2}}{21 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{3} x} - \frac {2 \, B}{7 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b x^{2}} + \frac {8 \, A c}{21 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2} x^{2}} - \frac {2 \, A}{9 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.41, size = 266, normalized size = 1.60 \begin {gather*} \frac {\sqrt {c\,x^2+b\,x}\,\left (x\,\left (\frac {4\,c^3\,\left (176\,A\,c-111\,B\,b\right )}{63\,b^5}+\frac {2\,c^3\,\left (247\,A\,c-138\,B\,b\right )}{63\,b^5}+\frac {b\,\left (\frac {184\,A\,c^5-96\,B\,b\,c^4}{63\,b^6}-\frac {4\,c^4\,\left (247\,A\,c-138\,B\,b\right )}{63\,b^6}\right )}{c}\right )+\frac {2\,c^2\,\left (176\,A\,c-111\,B\,b\right )}{63\,b^4}\right )}{x^2\,{\left (b+c\,x\right )}^2}-\frac {\sqrt {c\,x^2+b\,x}\,\left (18\,B\,b^3-52\,A\,b^2\,c\right )}{63\,b^6\,x^4}-\frac {\sqrt {c\,x^2+b\,x}\,\left (\frac {1024\,A\,c^4-768\,B\,b\,c^3}{63\,b^6}+\frac {2\,c\,x\,\left (1024\,A\,c^4-768\,B\,b\,c^3\right )}{63\,b^7}\right )}{x\,\left (b+c\,x\right )}-\frac {2\,A\,\sqrt {c\,x^2+b\,x}}{9\,b^3\,x^5}-\frac {2\,c\,\sqrt {c\,x^2+b\,x}\,\left (23\,A\,c-12\,B\,b\right )}{21\,b^5\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A + B x}{x^{3} \left (x \left (b + c x\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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